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Yes, such games exist.
There are many, actually! I've personally played at least two or three systems that match your description.
Among the thousands of RPGs out there, a good handful of them use no randomisation, or use randomisation other than dice (like drawing cards from a deck or tokens from a pouch).
Many, many RPGs --diced or not-- don't use ...

In the 5e DMG on page 261 there are
Noncombat Challenges
You use the DC required and the risk involved lieu of CR as a guideline to the difficulty level and then use the combat encounter guidelines to assign the exact amount of XP. The author recommend that XP only be awarded when there is a meaningful risk of failure. Note that risk does not have to mean ...

Just use combat rounds
This battle is against time. Have characters roll initiative and proceed in order, describing their actions according to the chart Actions in Combat. It's perfectly acceptable to explain to the players beforehand that this is not a combat encounter and that you're using combat mechanics to simulate a ticking clock, especially if ...

An answer to this question was posted on the Pokemon Tabletop adventures board by a user named Domo.
Based on what he's said, the main problems for the GM lies in the uselessness of Status effects because of the trainers actions, with only Burn or Poison remaining as an effective tool for the GM to combat Trainer interference with.
He also goes into action ...

Yes, games like that exist. There are many of them. You may wonder how such a game works, and how it differs from freeform improvisational collaborative storytelling.
To that end, I offer a modern, exemplary game of the type you are describing:
Hillfolk
The game system that drives Hillfolk is called the DramaSystem, by noted designer Robin D. Laws, and ...

All walls are passable, all mountains easily circumvented, all
mundane enemies easily slain, every weather condition easily changed,
every thought and alignment easily scryed, masses of people charmed at
no cost.
All of these things are extremely mundane. When you're dealing with Epic characters, you need a proper Epic setting, with proper Epic ...

Traditionally, experience comes from encounters, not necessarily combat. Much like real life, if you get yourself into a very difficult encounter with something or someone, then however it ended, you will probably walk away with some experience.
In other words, your group could choose to have a strong role-play game, with less focus on combat but where ...

Space Combat is still a combat encounter using Characteristic and Skill Tests -- just on a different playing field. The initiative is different and the question "what action do I take?" is brought to the foreground, but the core rules of how they are performed remains the same. During this combat, the character's personal characteristics are still being ...

Short answer: Nothing
After a read-through of the rules, it appears that there really isn't anything that needs to be done to run Pokemon Tabletop United in a closer-to-the-games style. Simply don't use any of the rules that pertain to a trainer's involvement in battles. This is backed up in the 1.05 Core PDF on page 7:
If you’re wanting to play a ...

This is a great question, and seems to me that many non-combat activities could be more fleshed out into more detailed game-able events. Usually things are left up to the GM, who sometimes might not be up to the task and think of everything as a flat skill roll. However there are some rules for various activities, or at least some guideline difficulty ...